Inter-track interference cancelation in the presence of frequency offset

ABSTRACT

An initial phase offset between a center track and a side track is determined. An initial side track pulse shape is determined using the initial phase offset and side track interference. The initial side track pulse shape minimizes a contribution of the side track interference to a center track bit. The contribution of the side track interference is removed from the center track bit using the initial side track pulse shape and the side track interference.

CROSS REFERENCE TO OTHER APPLICATIONS

This application claims priority to U.S. Provisional Patent ApplicationNo. 61/480,930 (Attorney Docket No. LINKP098+) entitled INTER-TRACKINTERFERENCE (ITI) CANCELLATION IN PRESENCE OF FREQUENCY OFFSET FORSHINGLED MAGNETIC RECORDING (SMR) filed Apr. 29, 2011, which isincorporated herein by reference for all purposes; this application isalso a continuation in part of co-pending U.S. patent application Ser.No. 13/282,370 (Attorney Docket No. LINKP074) entitled INTER-TRACKINTERFERENCE CANCELLATION FOR SHINGLED MAGNETIC RECORDING filed Oct. 26,2011, which is incorporated herein by reference for all purposes.

BACKGROUND OF THE INVENTION

Shingled magnetic recording (SMR) is a technique in which tracks areoverlapped. For example, a shingle is created by squeezing N trackstogether by making them overlap with each other. If a track overlapsanother track by δ, the increase in tracks per inch (TPI) TPI is givenby δ/(1−δ). This increase in TPI is attained at the cost of increasedinterference from side tracks as the read head will also sense orotherwise receive information from the side tracks in addition to thecenter track (sometimes referred to as the track of interest or thedesired track). If the additional information from the side tracks issignificant (e.g., because the side tracks are very close to the readhead, causing a relatively large amount of side track interference to bereceived), it may be necessary to remove the interference from the sidetrack(s) for successful decoding of the information bits. Although sometechniques exist to remove side track interference from a read backsignal, it would be desirable if improved techniques were developedwhich could be used in the presence of a frequency offset (e.g., anoffset between the write clock of a center track and the write clock ofa side track).

BRIEF DESCRIPTION OF THE DRAWINGS

Various embodiments of the invention are disclosed in the followingdetailed description and the accompanying drawings.

FIG. 1 is a diagram showing an embodiment of a center track and two sidetracks in a shingled magnetic recording (SMR).

FIG. 2 is a diagram showing an example of dividing a bit sequence intomultiple segments and performing a correlation-based correctiontechnique to each segment.

FIG. 3 is a diagram showing an embodiment of a phase offset after acorrection which is based on an adaptive technique.

FIG. 4 is a diagram showing an embodiment of a process for an adaptiveITI cancellation process.

FIG. 5 is a diagram showing an embodiment of a system for removing ITIusing a LMS adaptive technique.

FIG. 6 is a diagram showing example bit error rates using one embodimentof the proposed technique compared to some other techniques.

FIG. 7 is a diagram showing an embodiment of estimated pulse shapes fora side track.

DETAILED DESCRIPTION

The invention can be implemented in numerous ways, including as aprocess; an apparatus; a system; a composition of matter; a computerprogram product embodied on a computer readable storage medium; and/or aprocessor, such as a processor configured to execute instructions storedon and/or provided by a memory coupled to the processor. In thisspecification, these implementations, or any other form that theinvention may take, may be referred to as techniques. In general, theorder of the steps of disclosed processes may be altered within thescope of the invention. Unless stated otherwise, a component such as aprocessor or a memory described as being configured to perform a taskmay be implemented as a general component that is temporarily configuredto perform the task at a given time or a specific component that ismanufactured to perform the task. As used herein, the term ‘processor’refers to one or more devices, circuits, and/or processing coresconfigured to process data, such as computer program instructions.

A detailed description of one or more embodiments of the invention isprovided below along with accompanying figures that illustrate theprinciples of the invention. The invention is described in connectionwith such embodiments, but the invention is not limited to anyembodiment. The scope of the invention is limited only by the claims andthe invention encompasses numerous alternatives, modifications andequivalents. Numerous specific details are set forth in the followingdescription in order to provide a thorough understanding of theinvention. These details are provided for the purpose of example and theinvention may be practiced according to the claims without some or allof these specific details. For the purpose of clarity, technicalmaterial that is known in the technical fields related to the inventionhas not been described in detail so that the invention is notunnecessarily obscured.

FIG. 1 is a diagram showing an embodiment of a center track and two sidetracks in a shingled magnetic recording (SMR). Here, it is assumed thatthe center track may not be phase-aligned or frequency coherent (i.e.written with the same frequency relative to the media) with the sidetracks. In the example shown, diagram 100 shows three tracks in a SMRsystem. Track N (104) in this example is the center track (sometimesalso referred to as a desired track or track of interest). For example,data stored in track N (104) may be desired by some driver or otherhigher-level entity, whereas the data stored in the other tracks (i.e.,tracks 102 and 106) is not necessarily desired, at least at this pointin time.

In a SMR system, a read head (108) unintentionally senses the signalfrom the side tracks while reading the center track, causing the readback signal to include information from the side tracks in addition tothe center track. See, e.g., diagram 100, where the size of read head108 and the width and placements of tracks 102, 104, and 106 cause readhead 108 to be over at least a portion of side tracks 102 and 106. Theread head equalized output y_(k) ^(N) for center track 104 is given by:

$\begin{matrix}{y_{k}^{N} = {{\sum\limits_{i = {- L}}^{L}{h_{i}^{N}b_{k - i}^{N}}} + {\sum\limits_{i = {- L}}^{L}{h_{i}^{N - 1}b_{k - i - L^{N - 1}}^{N - 1}}} + {\sum\limits_{i = {- L}}^{L}{h_{i}^{N + 1}b_{k - i - L^{N + 1}}^{N + 1}}}}} & (1)\end{matrix}$

where h_(i) ^(N−1), h_(i) ^(N), h_(i) ^(N+1) are the equalized channelresponse for (N−1)^(th), N^(th) and (N+1)^(th) track with the read-headpositioned over track N, respectively, L^(N−1) and L^(N+1) are therelative phase mis-alignments of the N^(th) track to the (N−1)^(th) and(N+1)^(th) track, respectively, b_(k) ^(N) is the bit sequence for theN^(th) track, and L is length of the equalized channel response for(N−1)^(th), N^(th), and (N+1)^(th) tracks. In ITI cancellation, theseinterference terms from the side tracks (i.e., the second and thirdsummations, Σ_(i=−L) ^(L)h_(i) ^(N−1)b_(k−i−L) _(N−) ^(N−1) and Σ_(i=−L)^(L)h_(i) ^(N+1)b_(k−i−L) _(N+1) ^(N+1)) are removed from the read backsignal for improved performance, for example if there is too much sidetrack interference and downstream error correction decoders are not ableto properly decode the data.

To more clearly illustrate some of the terms in Equation (1), considerthe example of diagram 150. In that example, L^(N−1) represents thephase mis-alignment between tracks 152 and 154 at the beginning of therespective first bits in those tracks, and L^(N+1) represents the phasemis-alignment between tracks 154 and 156 at the beginning of therespective first bits in those tracks. In embodiments described herein,L^(N−1) and L^(N+1) are (e.g., the closest) integer values and in theexample of diagram 150, L^(N−1)=0 and L^(N+1)=1. In embodimentsdescribed herein, the values of L are integers and phase shift isspecified relative to the center track bit sequence; in some otherembodiments, some other (e.g., non-integer) values of L are permittedand/or the L is defined from some other point of reference. The valuesof L^(N−1), L^(N+1), and L determine which bits in the side tracksaffect the read head equalized output (y_(k) ^(N)) for the center track,at least as represented or otherwise modeled by Equation (1). In diagram150, for example, where L^(N−1)=0, L^(N+1)=1, and the length of theequalized channel response is 5, bits 4-8 of track (N−1) (i.e.,b_(k−i−L) _(N−1) ^(N−1) in the second summation of Equation (1)) andbits 5-9 of track (N+1) (i.e., b_(k−i−L) _(N+1) ^(N+)1 in the thirdsummation of Equation (1)) affect bit 6 in center track 154. The degreeto which these side track bits are sensed by a read head and included ina read back signal is reflected in or otherwise modeled by therespective side track pulse shapes, h_(i) ^(N−1) and h_(i) ^(N+1).

Diagram 150 also shows how a frequency offset affects the alignment ofbits in a side track compared to bits in a center track. At thebeginning of the bit sequence, the side track N+1 (156) starts first,followed by the center track (154) with the N−1 side track (152) last.However, because of differences in the write clock, at the end of thesequence when the same number of bits have been written (i.e., 10 bits),the N+1 side track (156) ends first, barely edging out the N−1 sidetrack (152) with the center track (154) ending last. In other words, thewrite clock of the N−1 side track is faster than that of the centertrack (154) (e.g., because even though it starts later it finishesfirst). Although not as noticeable as the difference between the centertrack and the N−1 side track, the write clock of the N+1 side track isslightly slower than that of the center track.

If there is frequency offset present in the side tracks, the channelpulse shapes for the side tracks will change with time. The read headequalized output y_(k) ^(N) for N^(th) center track in presence offrequency offset is given by:

$\begin{matrix}{y_{k}^{N} = {{\sum\limits_{i = {- L}}^{L}{h_{i}^{N}b_{k - i}^{N}}} + {\sum\limits_{i = {- L}}^{L}{{h_{i}^{N - 1}(k)}b_{k - i - L^{N - 1}}^{N - 1}}} + {\sum\limits_{i = {- L}}^{L}{{h_{i}^{N + 1}(k)}b_{k - i - L^{N + 1}}^{N + 1}}}}} & (2)\end{matrix}$

where h_(i) ^(N+1)(k) and h_(i) ^(N−1) (k) become time dependent unlikethe previous channel model. In other words, the pulse shapes h_(i)^(N+1)(k) and h_(i) ^(N−1)(k) change over time when there is a frequencyoffset; an ITI cancellation technique which is capable of operating insuch an environment would be desirable.

FIG. 2 is a diagram showing an example of dividing a bit sequence intomultiple segments and performing a correlation-based correctiontechnique to each segment. One technique for dealing with a frequencyoffset (described in co-pending U.S. patent application Ser. No.13/282,370, entitled INTER-TRACK INTERFERENCE CANCELLATION FOR SHINGLEDMAGNETIC RECORDING and filed on Oct. 26, 2011), is to divide a bitsequence into multiple bit sequences and apply the correlation techniqueto each smaller bit sequence. In this particular example, the bitsequence is divided into two segments.

Diagram 200 shows a phase offset before correction using thiscorrelation based technique; line 202 shows a linear phase offset whichresults from a (e.g., constant) frequency offset. Localized orinstantaneous noise may contribute a random additional phase offset toline 202, resulting in phase offset 204.

Diagram 250 shows the phase offset after correction of the first andsecond segments. Since a single correction is applied to each segment,correction tends to be best at the center of each segment and worst atthe edges of the segments. Lines 252 a and 252 b correspond to line 202and phase offsets 254 a and 254 b correspond to phase offset 204.

As shown in diagram 250, this technique (when used in systems with afrequency offset) may be undesirable for a variety of reasons. Noiseremoval may better with a longer bit sequence (and thus dividing a bitsequence into segments may comprise noise removal), performance may beless than optimal when segments are defined ahead of time because somedivision other than the pre-defined one may produce better results (buton the other hand having segments determined on-the-fly may becomputationally quite expensive), there may be relatively largediscontinuities or shifts at the boundaries of the segments once theside track interference is removed, removal of the side trackinterference is best in the middle of the bit sequences as opposed tonear the segment boundaries, and so on. It would therefore be desirableif new techniques could be developed which are (for example) relativelyeasy to implement in hardware and/or have better performance in thepresence of a frequency shift.

FIG. 3 is a diagram showing an embodiment of a phase offset after acorrection which is based on an adaptive technique. In the exampleshown, the adaptive technique does not divide a bit sequence intomultiple segments, but rather adjusts pulse shapes (i.e., h_(i) ^(N−1)and h_(i) ^(N+1)) in order to minimize a figure-of-merit (e.g., whichincludes or represents the side track interference as modified by thepulse shapes). Post-correction line 302 corresponds to pre-correctionline 202 in FIG. 2 and post-correction phase offset 304 corresponds topre-correction phase offset in 204 in FIG. 2.

A figure-of-merit is described herein which may be used for removing theinterference from one or both side tracks where there may be frequencyoffset present in either or both of the side tracks. In general, thefigure-of-merit includes or otherwise reflects the contributions fromthe side tracks when attempting to read the center track, as affected orotherwise filtered by the channel. In some embodiments, to removecontributions from the side tracks, the mean square error term isminimized with respect to h_(i) ^(N−1) and h_(i) ^(N+1). Put anotherway, the process attempts to determine values for h_(i) ^(N−1) and/orh_(i) ^(N+1) which, when used to construct an estimate of the sidetrackinterference terms, does so as accurately (in terms of mean squarederror) as possible. By determining the values of h_(i) ^(N−1) and/orh_(i) ^(N+1) in this manner, the side track interference term(s) can beconstructed accurately such that its removal from the center trackwaveform removes as much of this term as possible.

FIG. 4 is a diagram showing an embodiment of a process for an adaptiveITI cancellation process. For clarity, the example process describescancellation of ITI from a single side track but the technique may beexpanded to remove ITI from multiple side tracks (e.g., because sidetracks are independent and do not affect one other, similarly, removalof different side tracks are independent processes). Typically, the twoside tracks for a given center track have different pulse shapes andphase offset information; the example process shown below may beperformed on both tracks to determine the possibly different phaseoffsets and possibly different pulse shapes.

At 400, an initial phase offset between a center track and a side trackis determined. For example, in FIG. 2, an initial phase offset betweenside track N−1 (152) and center track N (154) may be determined at 400.Or, the initial phase offset between side track N+1 (156) and centertrack N (154) in FIG. 2 is determined at 400. In various embodiments,the number of bits in the side track and/or the number of bits in thecenter track which are used in estimating or calculating an initialphase offset is pre-defined, determined heuristically, etc.

At 402, an initial side track pulse shape is determined, using theinitial phase offset and side track interference, which minimizes thecontribution of side track interference to a current center track bit.For example, referring back to the FIG. 2, a side track pulse for trackN−1 (152) which minimizes the contribution of interference from thatside track is determined. Since the side track data for track N−1 (152)is known, the pulse shape which minimizes the contribution of that sidetrack interference can be determined.

In various embodiments, a variety of techniques may be used to performsteps 400 and 402. In some embodiments, a default or pre-defined valueis used. In some embodiments, an estimate or calculation is performed toobtain an initial phase offset and/or an initial side track pulse shape.In some embodiments it is assumed that an initial phase offset is zero(i.e., the adjacent tracks are assumed to be phase aligned and theinitial phase offset is set to zero). In some embodiments, an estimateor calculation used to determine an initial phase offset and/or aninitial side track pulse shape assumes that there is no frequency offsetbetween the center track and side track. This may be an acceptableassumption because the data used for estimating the phase offset andside track pulse shape may often and/or typically does not contain muchfrequency offset. For example, a 10 ppm frequency offset implies thatthere will be a 1 T shift in pulse shape in 100,000 T data samples. If5000 data samples are used for estimating pulse shape and phase offsetthen there will be a 0.05 T pulse shape shift which is negligible.

In some embodiments, determining an initial phase offset at 400 and/oran initial side track pulse shape at 402 includes one or more of thetechniques described in co-pending U.S. patent application Ser. No.13/282,370, entitled INTER-TRACK INTERFERENCE CANCELLATION FOR SHINGLEDMAGNETIC RECORDING and filed on Oct. 26, 2011. That applicationdescribes a technique in which inter-track interference (ITI) isdetected and canceled using correlation. It is assumed in that patentapplication that the side tracks b ^(N−1) and b ^(N+1) are decodable andare known. In at least some embodiments described therein, thecorrelation metric for the (N−1)^(th) track is defined as:

$\begin{matrix}{C_{m}^{N - 1} = {\sum\limits_{k = 1}^{T}{y_{k}^{N}b_{k - m}^{N - 1}}}} & (3)\end{matrix}$

where T is the number of samples used for the correlator. The values ofC_(m) ^(N−1) and C_(m) ^(N+1) may be computed for different values of mlying between −L_(max) to L_(max) where the value of L_(max) may bechosen heuristically. In some embodiments, an estimate of h^(N−1) isgiven by:

{circumflex over (h)} ^(N−) =[C _({circumflex over (m)}) _(N−1) ^(N−1) .. . C _({circumflex over (m)}) _(N−1) _(+L) ^(N−1])  (4)

where {circumflex over (m)}^(N−1)=arg max_(m)s_(m) ^(N−1), s_(m)^(N−1)=Σ_(i=0) ^(L)|C_(i+m) ^(N−1)|, and L is the length of h ^(N−1)which can be chosen heuristically. The estimate of L^(N−1) is given by

${\hat{L}}^{N - 1} = {{\hat{m}}^{N - 1} + \frac{L}{2} + 1.}$

The estimates for L^(N+1) and h ^(N+1) can be found in a similar manner.

In some embodiments, a least mean square (LMS) technique is used at 400to determine an initial phase offset and/or at 402 to determine aninitial side track pulse shape (e.g., as an alternative to thecorrelation-based technique described above). In LMS:

ĥ ^(LMS) ^(N−1) =arg min _(h) _(N−1) E[|e _(k) ¹|²], {circumflex over(h)} ^(LMS) ^(N+1) =arg min _(h) ^(N+1) E|e _(k) ²|²  (5)

where e_(k) ¹=y_(k) ^(N)−Σ_(i=−L) _(max) ^(L) ^(max) h_(i) ^(N−1)b_(k−i)^(N−1)b_(k−i) ^(N−1), e_(k) ²=y_(k) ^(N)−Σ_(i=−L) _(max) ^(L) ^(max)h_(i) ^(N+1)b_(k−i) ^(N+1), and L_(max) is the length of pulse shape(e.g., which is chosen heuristically). This problem can be solvedthrough recursion and the recursion equations are given by:

{circumflex over (h)} _(k+1) ^(LMS) ^(N−1) = {circumflex over (h)} _(k)^(LMS) ^(N−1) +μ₃ ×e _(k) ³ × b _(k) ^(N−1), {circumflex over (h)}_(k+1) ^(LMS) ^(N+1) = {circumflex over (h)} _(k) ^(LMS) ^(N+1) +μ₄ ×e_(k) ⁴ × b _(k) ^(N+1)  (6)

where μ₁ and μ₂ are adaptation coefficients, b _(k) ^(N−1)=[b_(k−L)_(max) ^(N−1) . . . b_(k+L) _(max) ^(N−1)] and b _(k) ^(N+1)=[b_(k−L)_(max) ^(N+1) . . . b_(k+L) _(max) ^(N+1)]. The adaptation coefficientsare chosen appropriately to ensure the estimates of the pulse shapeconverge to the correct solution. The phase mis-alignment for (N−1)^(th)track can be found through the converged recursion solution

${{\hat{\overset{\_}{h}}}^{{LMS}_{N - 1}}\mspace{14mu} {where}\mspace{14mu} {\hat{L}}^{N - 1}} = {\frac{L_{\max}}{2} + 1 - i_{\max}}$

where i_(max)=arg max_(i)h_(i)ĥ_(i) ^(LMS) ^(N−1) . The estimate for thephase mis-alignment for (N+1)^(th) track can be found in a similarmanner.

Returning back to FIG. 4, at 403, the contribution of the side trackinterference is removed from the current center track bit using the sidetrack pulse shape and side track data. For example, in the embodimentsassociated with Equations (7) and (8), this may include y_(k)^(N)−Σ_(i=−L) ^(L)h_(i) ^(N−1)b_(k−i−{circumflex over (L)}) _(N−1)^(N−1) for removing interference associated with the (N−1)th side trackand/or y_(k) ^(n)−Σ_(i=−L) ^(L)h_(i) ^(N+1)b_(k−i−{circumflex over (L)})_(N+1) ^(N+1) for removing interference associated with the (N+1)th sidetrack.

At 404, it is determined whether to continue the process. In someembodiments, for example, a side track pulse shape is determined foreach bit in a center track, and the process continues so long as thereare bits remaining in the center track. If the process continues, at406, a next side track pulse shape is determined, using a previous sidetrack pulse shape and side track data, which minimizes contribution ofside track interference to current center track bit. In someembodiments, an LMS based technique is used.

If it is determined for the process to continue, at 406 a next sidetrack pulse shape is determined, using previous side track pulse shapeand side track interference, which minimizes the contribution of theside track interference to a current center track bit.

In some embodiments, the center track data is estimated to a reliable orsufficient degree, and a LMS process at step 406 operates on afigure-of-merit from which the (estimated) center track information isremoved. In this embodiment, the estimates {circumflex over (b)} ^(N) ofthe information bits are present on the estimator side. It is alsoassumed that a significant amount of frequency offset is present whichwill make the pulse shape coefficients h_(i) ^(N+1)(k) and h_(i)^(N−1)(k) time dependent. The LMS estimator recursion equation isadaptive over time and can be used in the presence of a frequencyoffset. The LMS estimator recursion equations are given by:

{circumflex over (h)} _(k+1) ^(LMS) ^(N−1) = {circumflex over (h)} _(k)^(LMS) ^(N−1) +μ₃ ×e _(k) ³ × b _(k) ^(N−1), {circumflex over (h)}_(k+1) ^(LMS) ^(N+1) = {circumflex over (h)} _(k) ^(LMS) ^(N+1) +μ₄ ×e_(k) ⁴ × b _(k) ^(N+1)  (7)

where e_(k) ³=y_(k) ^(N)−Σ_(i=−L) ^(L)h_(i) ^(N){circumflex over(b)}_(k−i) ^(N)−Σ_(i=−L) ^(L)h_(i) ^(N−1)b_(k−i−{circumflex over (L)})_(N−1) ^(N−1), e_(k) ⁴=y_(k) ^(N)−Σ_(i=−L) ^(L)h_(i) ^(N){circumflexover (b)}_(k−i) ^(N)−Σ_(i=−L) ^(L)h_(i)^(N+1)b_(k−i−{circumflex over (L)}) _(N+1) ^(N+1), μ₃ and μ₄ are the LMSadaptation coefficients. These recursion equations give the pulse shapeestimates at each time instant and may be used to remove ITI from thecenter track.

In some cases, the estimates of the center track information ({circumflex over (b)} ^(N)) are not available. Or, the signal to noiseratio (SNR) for that information is relatively low, which means that theestimates {circumflex over (b)} ^(N) are noisy and it would not bedesirable to remove the center track information. In such cases, the LMSestimator recursion equations are given by:

{circumflex over (h)} _(k+1) ^(LMS) ^(N−1) = {circumflex over (h)} _(k)^(LMS) ^(N−1) +μ₅ ×e _(k) ⁵ × b _(k) ^(N−1), {circumflex over (h)}_(k+1) ^(LMS) ^(N+1) = {circumflex over (h)} _(k) ^(LMS) ^(N+1) +μ₆ ×e_(k) ⁶ × b _(k) ^(N+1)  (8)

where e_(k) ⁵=y_(k) ^(N)−Σ_(i=−L) ^(L)h_(i)^(N−1)b_(k−i−{circumflex over (L)}) _(N−1) ^(N−1), e_(k) ⁶=y_(k)^(N)−Σ_(i=−L) ^(L)h_(i) ^(N+1)b_(k−i−{circumflex over (L)}) _(N+1)^(N+1), μ₅ and μ₆ are the LMS adaptation coefficients. Note that, unlikethe equations for e_(k) ³ and e_(k) ⁴, the equations for e_(k) ⁵ ande_(k) ⁶ do not remove the term Σ_(i=−L) ^(L)h_(i) ^(N){circumflex over(b)}_(k−i) ^(N) from the center track information (y_(k) ^(N)). Asdescribed above, in some embodiments, gradients are used to determinethe side track pulse shapes which minimize e_(k) ⁵ and e_(k) ⁶. As inthe previous example, the pulse shape estimates at each time instant maybe used to remove the ITI from the side track(s) from the center track.There are various other ways in which the LMS estimator problem can beformulated (e.g., first the pulse shape for the (N−1)^(th) track can bedetected and further used to remove its interference effect whiledetecting the pulse shape for (N+1)^(th) track).

FIG. 5 is a diagram showing an embodiment of a system for removing ITIusing a LMS adaptive technique. In the example shown, the LMS system isincluded in a SMR system. For clarity, other components (e.g.,associated with writing to the magnetic storage media) are not shown inthis figure.

A signal (y_(k) ^(N)) which includes both center track signal andinterference from two side tracks is input to the system, as are the bitsequences associated with the two side tracks (i.e., b ^(N−1) and b^(N+1)). Initial phase offset estimators 500 and 502 output an initialphase offset for the (N−1)^(th) side track and the (N+1)^(th) sidetrack. These initial phase offsets (i.e., {circumflex over (L)}^(N−1)and {circumflex over (L)}^(N+1)) are passed to adjustable delay elements504 and 506). Delay elements 504 and 506 delay the bit sequence of the(N−1)^(th) side track and the (N+1)^(th) side track, respectively, bythe specified amount so that the delayed versions of the side tracks arerelatively lined up with the y_(k) ^(N) signal (e.g., it may not benecessary to have the side tracks exactly line up with y_(k) ^(N);“close enough” may be sufficient).

LMS side track pulse estimators 508 and 510 received the delayed sidetrack bit sequences as well as the y_(k) ^(N) signal. Using least meansquares (LMS), side track pulse shapes for the respective side tracksare generated for the (N−1)¹¹¹ side track and the (N+1)^(1h) side trackby estimators 508 and 510, respectively. For example, estimators 508 and510 may use the LMS estimator recursion equations shown in Equation (7)(e.g., when the center track information is known and/or has asufficient SNR) or in Equation (8) (e.g., when the center trackinformation is not known, or it does not have a sufficiently high SNR).Alternatively, some other embodiments use non-LMS side track pulseestimation techniques to obtain the pulse shapes. In some embodiments, anew pulse shape is generated for each bit of the signal y_(k) ^(N). Inother words, as each bit of the signal y_(k) ^(N) is input, new sidetrack pulse shapes are generated with each bit that is input.

The side track pulse shapes for the side tracks are combined with theirrespective, delayed bit sequence and are then removed from the y_(k)^(N) signal. Ideally, the side channel bit sequences used by the system(i.e., b ^(N−1) and b ^(N+1)) perfectly match actual side channel bitsequences and the estimated side track pulse shapes (i.e., ĥ _(k) ^(LMS)^(N−1) and ĥ _(k) ^(LMS) ^(N+1) ) perfectly match the actual channel(e.g., based on the read arm positioning of the read arm, thepositioning of the side tracks, etc.) so that the output signal has noside track interference and only center track information.

One advantage, at least with ITI cancellation systems which use LMS, isthat they have relatively low complexity and can be relatively easilyimplemented in hardware. For example, storage controllers, such as SMRcontrollers, are often implemented on hardware (such as anapplication-specific integrated circuit (ASIC), a field-programmablegate array (FPGA), or a device with a microprocessor, such as an ARMcore) and therefore ease of hardware implementation is often animportant consideration.

FIG. 6 is a diagram showing example bit error rates using one embodimentof the proposed technique compared to some other techniques. In theexample shown, a SMR system with a channel model was simulated. In theexample simulation, a sector containing 40000 bits of data wasoverlapped with two side tracks. Different values of the frequencyoffset varying from 1 parts per million (ppm) to 800 ppm are used in thesimulation. Bit error rate 600 (line with circle data points) shows anerror rate for a system with no ITI cancellation. Bit error rate 602(line with triangular data points) shows an error rate for a system withsome other ITI cancellation technique. Bit error rate 604 (line withsquare data points) shows an error rate for one embodiment of the ITIcancellation technique described herein.

It is not uncommon for a system to have a frequency offset in the rangeof 100 ppm. At that frequency offset, the performance of the ITIcancellation technique (i.e., a BER of 0.06) is better than that of theother ITI cancellation technique (i.e., a BER of ˜0.16) and the systemwith no ITI cancellation (i.e., a BER of ˜1.9). Even when the frequencyoffset is significant (e.g., 800 ppm), the performance of the embodimentused in the simulation is better than that of the other ITI cancellationtechnique.

FIG. 7 is a diagram showing an embodiment of estimated pulse shapes fora side track. In this example, the estimated pulse shapes were obtainedby simulating a 400 ppm frequency offset present in both side tracks. Asa result of the frequency offset, the pulse shapes change over time. Inthe example shown, the pulse shapes at 16 different points in time gofrom left to right and from top to bottom (i.e., pulse shape 700corresponds to a first point in time or a first bit in a sequence, pulseshape 702 corresponds to a second point in time or a second bit in asequence, pulse shape 750 corresponds to a second to last point in timeor a second to last bit in a sequence, and pulse shape 752 correspondsto a last point in time or a last bit in a sequence). Referring back toEquation (7), pulse shape 700 is an example of ĥ ₀ ^(LMS) ^(N−1) or ĥ ₀^(LMS) ^(N+1) , pulse shape 702 is an example of ĥ ₁ ^(LMS) ^(N−1) or ĥ₁ ^(LMS) ^(N+1) , pulse shape 750 is an example of ĥ ₁₄ ^(LMS) ^(N−1) orĥ ₁₄ ^(LMS) ^(N+1) , and pulse shape 752 is an example ĥ ₁₄ ^(LMS)^(N+1) or ĥ ₁₅ ^(LMS) ^(N+1) . In some embodiments, pulse shapes aregenerated for each bit in a center track (e.g., in FIG. 1, side trackpulse shapes are generated for each of the 10 bits in center track 154,multiplied by two side tracks for a total of 20 (e.g., different) sidetrack pulse shapes generated.

Although the foregoing embodiments have been described in some detailfor purposes of clarity of understanding, the invention is not limitedto the details provided. There are many alternative ways of implementingthe invention. The disclosed embodiments are illustrative and notrestrictive.

1. A method, comprising: determining an initial phase offset between acenter track and a side track; using a processor to determine an initialside track pulse shape, using the initial phase offset and side trackinterference, wherein the initial side track pulse shape minimizes acontribution of the side track interference to a center track bit; andremoving the contribution of the side track interference from the centertrack bit using the initial side track pulse shape and the side trackinterference.
 2. The method of claim 1, wherein the center track and theside track are associated with a shingled magnetic recording (SMR)system.
 3. The method of claim 1, wherein determining the initial phaseoffset includes using${{\hat{L}}^{N - 1} = {{\hat{m}}^{N - 1} + \frac{L}{2} + 1}},$ where{circumflex over (m)}^(N−1)=arg max_(m)s_(m) ^(N−1) and s_(m)^(N−1)=Σ_(i=0) ^(L)|C_(i+m) ^(N−1)|.
 4. The method of claim 1, whereinusing the processor to determine the initial side track pulse shapeincludes using a correlation metric C_(m) ^(N−1)=Σ_(k=1) ^(T)y_(k)^(N)b_(k−m) ^(N−1), where T is a number of samples used for thecorrelation.
 5. The method of claim 1, wherein determining the initialphase offset includes using${{\hat{L}}^{N - 1} = {\frac{L_{\max}}{2} + 1 - i_{\max}}},$ wherei_(max)=arg max_(i)ĥ_(i) ^(LMS) ^(N−1) and L_(max) is the length of theinitial side track pulse shape.
 6. The method of claim 1, wherein usingthe processor to determine the initial side track pulse shape includesusing ĥ ^(LMS) ^(N−1) =arg min _(h) _(N−1) E[|e_(k) ¹|²], where e_(k)¹=y_(k) ^(N)−Σ_(i=−L) _(max) ^(L) ^(max) h_(i) ^(N−1)b_(k−i) ^(N−1) andL_(max) is the length of the initial side track pulse shape.
 7. Themethod of claim 1, wherein using the processor to determine the initialside track pulse shape includes using ĥ _(k+1) ^(LMS) ^(N−1) = ĥ _(k)^(LMS) ^(N−1) +μ₁×e_(k) ¹× b _(k) ^(N−1), where e_(k) ¹=y_(k)^(N)−Σ_(i=−L) _(max) ^(L) ^(max) h_(i) ^(N−1)b_(k−i) ^(N−1), L_(max) isthe length of the initial side track pulse shape, μ₁ is an adaptationcoefficient, and b _(k) ^(N−1)=[b_(k−L) _(max) ^(N−1) . . . b_(k+L)_(max) ^(N−1)].
 8. The method of claim 1, further comprising: using theprocessor to determine a second side track pulse shape, using theinitial side track pulse shape and the side track data, wherein thesecond side track pulse shape minimizes the contribution of the sidetrack interference to a second center track bit; and remove thecontribution of the side track interference from the second center trackbit using the second side track pulse shape and side track data.
 9. Themethod of claim 8, wherein using the processor to determine the secondside track pulse shape includes using ĥ _(k+1) ^(LMS) ^(N−1) = ĥ _(k)^(LMS) ^(N−1) +μ₅×e_(k) ⁵× b _(k) ^(N−1), where e_(k) ⁵=y_(k)^(N)−Σ_(i=−L) ^(L)h_(i) ^(N−1)b_(k−i−{circumflex over (L)}) _(N−1)^(N−1) is an adaptation coefficient.
 10. A system, comprising: aprocessor; and a memory coupled with the processor, wherein the memoryis configured to provide the processor with instructions which whenexecuted cause the processor to: determine an initial phase offsetbetween a center track and a side track; determine an initial side trackpulse shape, using the initial phase offset and side track interference,wherein the initial side track pulse shape minimizes a contribution ofthe side track interference to a center track bit; and remove thecontribution of the side track interference from the center track bitusing the initial side track pulse shape and the side trackinterference.
 11. The system of claim 10, wherein the system isassociated with shingled magnetic recording (SMR).
 12. The system ofclaim 10, wherein the instructions for determining the initial phaseoffset include instructions for using${{\hat{L}}^{N - 1} = {{\hat{m}}^{N - 1} + \frac{L}{2} + 1}},$ where{circumflex over (m)}^(N−1)=arg max_(m)s_(m) ^(N−1) and s_(m)^(N−1)=Σ_(i=0) ^(L)|C_(i+m) ^(N−1)|.
 13. The system of claim 10, whereinthe instructions for determining the initial side track pulse shapeinclude instructions for using a correlation metric C_(m) ^(N−1)=Σ_(k=1)^(T)y_(k) ^(N)b_(k−m) ^(N−1), where T is a number of samples used forthe correlation.
 14. The system of claim 10, wherein the instructionsfor determining the initial phase offset include instructions for using${{\hat{L}}^{N - 1} = {\frac{L_{\max}}{2} + 1 - i_{\max}}},$ wherei_(max)=arg max_(i)ĥ_(i) ^(LMS) ^(N−1) and L_(max) is the length of theinitial side track pulse shape.
 15. The system of claim 10, wherein theinstructions for determining the initial side track pulse shape includeinstructions for using ĥ ^(LMS) ^(N−1) =arg min _(h) _(N−1) E[|e_(k)¹|²], where e_(k) ¹=y_(k) ^(N)−Σ_(i=−L) _(max) ^(L) ^(max) h_(i)^(N−1)b_(k−i) ^(N−1) and L_(max) is the length of the initial side trackpulse shape.
 16. The system of claim 10, wherein the instructions fordetermining the initial side track pulse shape include instructions forusing ĥ _(k+1) ^(LMS) ^(N−1) = ĥ _(k) ^(LMS) ^(N−1) +μ₁×e_(k) ¹× b _(k)^(N−1), where e_(k) ¹=y_(k) ^(N)−Σ_(i=−L) _(max) ^(L) ^(max) h_(i)^(N−1)b_(k−i) ^(N−1), L_(max) is the length of the initial side trackpulse shape, μ₁ is an adaptation coefficient, and b _(K) ^(N−1)=[b_(k−L)_(max) ^(N−1) . . . b_(k+L) _(max) ^(N−1)].
 17. The system of claim 10,wherein the memory is further configured to provide the processor withinstructions which when executed cause the processor to: determine asecond side track pulse shape, using the initial side track pulse shapeand the side track interference, wherein the second side track pulseshape minimizes the contribution of the side track interference to asecond center track bit; and remove the contribution of the side trackinterference from the second center track bit using the second sidetrack pulse shape and side track data.
 18. The system of claim 17,wherein using the processor to determine the second side track pulseshape includes using ĥ _(k+1) ^(LMS) ^(N−1) = ĥ _(k) ^(LMS) ^(N−1)+μ₅×e_(k) ⁵× b _(k) ^(N−1), where e_(k) ⁵=y_(k) ^(N)−Σ_(i=−L) ^(L)h_(i)^(N−1)b_(k−i−{circumflex over (L)}) _(N−1) ^(N−1) and μ₅ is anadaptation coefficient.
 19. A computer program product, the computerprogram product being embodied in a tangible computer readable storagemedium and comprising computer instructions for: determining an initialphase offset between a center track and a side track; using a processorto determine an initial side track pulse shape, using the initial phaseoffset and side track interference, wherein the initial side track pulseshape minimizes a contribution of the side track interference to acenter track bit; and removing the contribution of the side trackinterference from the center track bit using the initial side trackpulse shape and the side track interference.
 20. The computer programproduct of claim 19, wherein the center track and the side track areassociated with a shingled magnetic recording (SMR) system.